Properties

Proposition 1:

$$\mbox{For any} X \mbox{ and } Y E(X+Y)=E(X)+E(Y)$$

Proposition 2:
If X is a continuous random variable with probability density function f(x) then for any real valued function g:

$$E[g(X)]=\int_{-\infty}^{\infty} g(x)f(x)dx$$

Proposition 3:

E(aX+b)=aE(X)+b

Proposition 4:
Only for independent random variables do we have

$$If\;\; X\; and \; Y\; independent\; E(XY)=E(X)E(Y)$$

Examples:

For the exponential random variable with parameter $\lambda$, I showed:

$$E[X]=\frac{1}{\lambda}$$

For U a random uniform on [0,1], I showed: $E(U)=\frac{1}{2}$.

$$Z \sim Normal(0,1) E(Z)=0$$

$$X \sim Normal(\mu,\sigma^2), E(X)=\mu$$